In this paper, a procedure known as dual decomposition is used to solve
the MMG operation model. This procedure has been widely applied
to various optimization problems that benefit from a distributed solution.
A generic description of dual decomposition, as presented in
[1]}, can be explained based on the following mathematical
model:
[2]
x i=1I fi(xi)  
gi(xi)Ai     i

i=1I hi(xi)B
   

where the objective function \(f_{i} (x_{i})\)  is a convex, quadratic, and
separable function with respect to variables \(x_{i}\) .
Constraints () are also separable (one constraint
for each \(i\) ), but constraints ()
are complicating constraints because they are not separable and prevent
the decomposition of the problem.
In order to relax the complicating constraints, a dual decomposition procedure
is used by dualizing (),
as follows [2]}:

xi i=1I fi(xi)
+ [ B - i=1I hi(xi) ]

max{
gi(xi)
Ai   
i }

This problem is not decomposable, due to the Lagrange multiplier (or dual
variable) \(\lambda \)  in the objective function. To solve this issue, the problem
is relaxed by fixing \(\lambda \)  to a constant value \(\overline{\lambda }\) , i.e.,
turning it into a parameter, with the relaxed problem being decomposable into
the following \(i\)  subproblems:

xifi(xi) -  hi(xi)

gi(xi)Ai
   
